Activity 29.
For each function given below, identify its fundamental algebraic structure. In particular, is the given function a sum, product, quotient, or composition of basic functions? If the function is a composition of basic functions, state a formula for the inner function \(g\) and the outer function \(f\) so that the overall composite function can be written in the form \(f(g(x))\text{.}\) If the function is a sum, product, or quotient of basic functions, use the appropriate rule to determine its derivative.
(a)
\(\displaystyle h(x) = \cos(e^x)\)
Answer.
This is a composition of functions. \(h(x)=f(g(x))\) where \(f(x)=\cos x\) and \(g(x)=e^x\text{.}\)
(b)
\(\displaystyle p(x) = x^3 e^x\)
Solution.
This is a product of functions. The product rule gives
\begin{align*}
p'(x) = \amp \mathstrut \frac{d}{dx}(x^3)e^x + x^3\frac{d}{dx}(e^x)\\
= \amp \mathstrut 3x^2\cdot e^x + x^3 \cdot e^x \\
= \amp \mathstrut x^2e^x (3+x).
\end{align*}
(c)
\(\displaystyle r(x) = (\cos x)^3\)
Answer.
This is a composition of functions. \(r(x)=f(g(x))\) where \(f(x) = x^3\) and \(g(x) = \cos x\text{.}\)
(d)
\(\displaystyle m(x) = 3^{\cos x}\)
Answer.
This is a composition of functions. \(m(x) = f(g(x))\) where \(f(x) = 3^x\) and \(g(x) = \cos x\text{.}\)
(e)
\(\displaystyle w(x) = \sqrt{\cos x}\)
Answer.
This is a composition of functions. \(w(x) = f(g(x))\) where \(f(x) = \sqrt{x}\) and \(g(x) = \cos x\text{.}\)
(f)
\(\displaystyle w(x) = \sqrt{\cos x} + 3^{\cos x}\)
Answer.
This is a composition of functions. \(w(x) = f(g(x))\) where \(f(x) = \sqrt{x} + 3^x\) and \(g(x) = \cos x\text{.}\)
(g)
\(\displaystyle s(x) = \cos x - \sqrt{x} - \frac{3^x}{\cos x}\)
Solution.
This is a difference of functions. The quotient rule can also be applied. \(s'(x) = -\sin x - \frac{1}{2}x^{-1/2} - \frac{\cos x \cdot (\ln 3) 3^x - 3^x (-\sin x)}{\cos^2 x}\text{.}\)
(h)
\(\displaystyle p(x) = 4\cos(\sqrt{x}) - 3^{\sqrt{x}}\)
Solution.
This is a composite of functions. \(p(x) = f(g(x))\) where \(f(x) = 4\cos x - 3^x\) and \(g(x)=\sqrt{x}\text{.}\)
(i)
\(\displaystyle r(x) = 4\cos(\sqrt{x}) - 3^{\sqrt{x}} + 7\)
Solution.
This is a composite of functions. \(r(x) = f(g(x))\) where \(f(x) = 4\cos x - 3^x+7\) and \(g(x)=\sqrt{x}\text{.}\)
(j)
\(\displaystyle r(x) = 4\cos(\sqrt{x}+2) - 3^{\sqrt{x}+2}\)
Solution.
This is a composite of functions. \(r(x) = f(g(x))\) where \(f(x) = 4\cos x - 3^x\) and \(g(x)=\sqrt{x}+2\text{.}\)
(k)
The two functions that follow are each a composition of three basic functions. State formulas for \(f\text{,}\) \(g\text{,}\) and \(h\) so that the function can be written as \(f(g(h(x)))\text{.}\)
\(\displaystyle p(x) = 3^{\cos(\sqrt{x})}\)
Solution.
This is a composite of three fundamental functions. \(p(x) = f(g(h(x)))\) where \(f(x) = 3^x\text{,}\) \(g(x)=\cos x\text{,}\) and \(h(x) = \sqrt{x}\text{.}\)
(l)
\(\displaystyle p(x) = 2(\sqrt{\cos x})^3 + 3^{\sqrt{\cos x}}\)
Solution.
This is a composite of three fundamental functions. \(r(x) = f(g(h(x)))\) where \(f(x) = 2x^3 + 3^x\text{,}\) \(g(x)=\sqrt{x}\text{,}\) and \(h(x) = \cos x\text{.}\)
